Iterative and Algebraic Algorithms for the Computation of the Steady State Kalman Filter Gain

The Kalman filter gain arises in linear estimation and is associated with linear systems. The gain is a matrix through which the estimation and the prediction of the state as well as the corresponding estimation and prediction error covariance matrices are computed. For time invariant and asymptotically stable systems, there exists a steady state value of the Kalman filter gain. The steady state Kalman filter gain is usually derived via the steady state prediction error covariance by first solving the corresponding Riccati equation. In this paper, we present iterative per-step and doubling algorithms as well as an algebraic algorithm for the steady state Kalman filter gain computation. These algorithms hold under conditions concerning the system parameters. The advantage of these algorithms is the autonomous computation of the steady state Kalman filter gain.


Introduction
The Kalman filter gain arises in Kalman filter equations in linear estimation and is associated with linear systems.State space systems have been widely used in estimation theory to describe discrete time systems [1][2][3][4][5].It is known [1] for time invariant systems that if the signal process model is asymptotically stable, then there exists a steady state value of the Kalman filter gain.Thus, the steady state gain is associated with time invariant systems described by the following state space equations: for  ≥ 0, where   is the -dimensional state vector at time ,   is the -dimensional measurement vector at time ,  is the × system transition matrix,  is the × output matrix,   is the plant noise at time , and V  is the measurement noise at time .Also, {  } and {V  } are Gaussian zero-mean white random processes with covariance matrices  and , respectively.
The discrete time Kalman filter [1,6] is the most wellknown algorithm that solves the filtering problem.In fact, Kalman filter faces simultaneously two problems as follows.
(i) Estimation: the aim is to recover at time  information about the state vector at time  using measurements up till time .
(ii) Prediction: the aim is to obtain at time  information about the state vector at time  + 1 using measurements up till time ; it is clear that prediction is related to the forecasting side of information processing.
Kalman filter uses the measurements up till time  in order to produce the (one step) prediction of the state vector and the corresponding prediction error covariance matrix  +1/ , as well as producing the estimation of the state vector and the corresponding estimation error covariance matrix  / .The Kalman filter equations needed for the computation of 2 ISRN Applied Mathematics the prediction and estimation error covariance matrices are as follows: for  ≥ 0, with initial condition  0/−1 =  0 for the time instant, where there are no measurements given.Note that   is the Kalman filter gain.From ( 2) to (4), we are able to derive the Riccati equation, which is an iterative equation with respect to the prediction error covariance: In the general case, where  and  0 are positive definite matrices, using in (5) the matrix inversion lemma: the Riccati equation is formulated as The Riccati equation is a nonlinear iterative equation with respect to the prediction error covariance.For time invariant systems, it is well known [1] that if the signal process model is asymptotically stable, then there exists a steady state value P of the prediction error covariance matrix.In fact, the prediction error covariance tends to the steady state prediction error covariance.
The steady state prediction error covariance satisfies the steady state Riccati equation Then, from (2), it is clear that there also exists a steady state value K of the Kalman filter gain [7].The steady state gain can be calculated by Also, from (3), it is clear that there also exists a steady state value P of the estimation error covariance matrix [7], which can be calculated by It is obvious from (9) that the steady state Kalman filter gain can be derived via the steady state prediction error covariance.The covariance matrix in Kalman filter plays an important role in many applications [1,4,6,[8][9][10].The steady state prediction error covariance can be derived by solving the Riccati equation emanating from Kalman filter.The discrete time Riccati equation has attracted recent attention.In view of the importance of the Riccati equation, there exists considerable literature on its algebraic solutions; for example, in [1,7,11,12], the authors have derived an eigenvector solution, while the author of [13] has included solving scalar polynomials.Other methods are based on the iterative solutions [1,[13][14][15][16][17][18] concerning per-step or doubling algorithms.The iterative algorithms that provide the steady state Kalman filter gain together with the prediction error covariance are the Chandrasekhar algorithms [1], as well as the iterative algorithm that calculate the Kalman gain only once for a period of the stationary channel, as opposed to each data sample in the conventional filter [19].A geometric illustration of the Kalman filter gain is given in [20].
In this paper, we present algorithms for the steady state Kalman filter gain autonomous computation.These algorithms hold under conditions concerning the system parameters.The paper is organized as follows: two new perstep iterative algorithms, a new doubling iterative algorithm, and an algebraic algorithm for the computation of the steady state Kalman filter gain are presented in Section 2. In Section 3, two examples verify the results.Finally, Section 4 summarizes the conclusions.

New Algorithms for the Steady State
Kalman Filter Computation 2.1.Assumptions.We assume the general case, where  and  0 are positive definite matrices.The Kalman filter gain   is a matrix of dimension  × .
We define the matrix It is clear that   is a nonsymmetric matrix of dimension ×.
It is also clear that there exists a steady state value Also, we define the matrix Note that  is an  ×  symmetric positive semidefinite matrix and  is a positive definite if rank() = ; this means that  is a nonsingular matrix in the case, rank() =  with  ≥ , [21].

Indirect Steady State Kalman Filter Gain Computation.
In this section, we present algorithms for G computation.Then, we show how to compute the steady state Kalman filter K through G = K.

Iterative Algorithms for G Computation.
In this section, we present two iterative per-step algorithms and an iterative doubling algorithm for G computation.
Per-Step Iterative Algorithm 1.Using ( 2) and ( 11), it is derived that Thus, arises Using the Riccati equation ( 7), (15), the nonsingularity of , and some algebra we have Also, from ( 2) and ( 13), we can write Since the matrices  /−1 ,  are nonsingular, the last equation yields Substituting in ( 16) the matrix  +1  −1  −1 +1/ by ( 18), it follows whereby it is implied that Thus, the above equation can be written as Combining ( 21) with (11), the following nonlinear iterative equation with respect to   is derived: where The algorithm uses the initial condition  0 =  0  =  0   [ 0   + ] −1 .It is known [1] that the prediction error covariance tends to the steady state prediction error covariance and that the convergence is independent of the initial uncertainty, that is, independent of the value of the initial condition  0 .Thus, we are able to assume zero initial condition  0 = 0 and so we are to use the initial condition  0 = 0.
It is clear that   tends to a steady state value G and by (22 Per-Step Iterative Algorithm 2. We rewrite (22) as Thus, the following nonlinear iterative equation with respect to   is derived: where The algorithm uses the initial condition  0 =  0  =  0   [ 0   + ] −1 .It is known [1] that the prediction error covariance tends to the steady state prediction error covariance and that the convergence is independent of the initial uncertainty, that is, independent of the value of the initial condition  0 .Thus, we are able to assume zero initial condition  0 = 0.In this case, in order to avoid  −1 0 , we are to use the initial condition  1 = .
It is clear that   tends to a steady state value G and by (26) G satisfies Doubling Iterative Algorithm.Ιn (22), setting we take or where is a matrix of dimension 2 × 2 and , , ,  as in (23).
We are able to use zero initial condition  0 = 0, so  0 =  0  =  0  −1 0 = 0; that is, and hence We define with initial condition Then, and, using the doubling principle Then we are able to derive, after some algebra, the following nonlinear iterative equations: with initial conditions Then, since it is clear that   =  2   −1 2  =  2  tends to a steady state value G.

Algebraic Algorithm for G Computation.
In this section, we present an algebraic algorithm for G computation.As in (29), setting and using the parameters , , ,  by (23), we derive which is a matrix of dimension 2 × 2.Since it is evident that Φ is a nonsingular matrix and its eigenvalues occur in reciprocal pairs.Thus, (43) can be written where is a diagonal matrix containing the eigenvalues of Φ, with Λ diagonal matrix with all the eigenvalues of Φ lying outside the unit circle, and is the matrix containing the corresponding eigenvectors of Φ, with We are able to use zero initial condition  0 = 0, so  0 =  0  =  0  −1 0 = 0; that is, and hence Then, from ( 50) and ( 45)-( 48), we are able to write that is, Substituting in (42) the matrices   ,   from (52), we have that Furthermore, the diagonal matrix Λ −1 contains all the eigenvalues of Φ lying inside the unit circle, which follows that lim  → ∞ Λ − = 0.Then,   tends to a steady state value G with G = lim  → ∞   , and from (53) arises

Direct Steady State Kalman Filter Gain Computation.
In this section, we present algorithms for the direct computation of the steady state Kalman filter K.The proposed algorithms compute directly the steady state Kalman filter gain, that is, without using G = K.All these algorithms hold under the assumption that  = .Note that, since rank() = ,  and  are nonsingular matrices.

Iterative Algorithms for K Computation.
In this section, we present two iterative per-step algorithms and an iterative doubling algorithm for K computation.
Per-Step Iterative Algorithm 1.Using ( 11), (22), and ( 13), we are able to derive the following nonlinear iterative equation with respect to the Kalman filter gain   : The nonsingularity of  and (13) allow us to write the equality in (56) as where The initial condition is  0 =  0   [ 0   + ] −1 .It is known [1] that the prediction error covariance tends to the steady state prediction error covariance and that the convergence is independent of the initial uncertainty, that is, independent of the value of the initial condition  0 .Thus, we are able to assume zero initial condition  0 = 0 and so we are to use the initial condition  0 = 0.
It is clear that   tends to a steady state value K satisfying Per-Step Iterative Algorithm 2. Using (57), we are able to derive the following nonlinear iterative equation with respect to the Kalman filter gain   : where , , ,  are given by (58) and The algorithm uses the initial condition  0 =  0   [ 0   + ] −1 .It is known [1] that the prediction error covariance tends to the steady state prediction error covariance and that the convergence is independent of the initial uncertainty, that is, independent of the value of the initial condition  0 .Thus, we are able to assume zero initial condition  0 = 0.In this case, in order to avoid  −1 0 , we are to use the initial condition  1 = .
It is clear that   tends to a steady state value K satisfying Doubling Iterative Algorithm.Ιn (57), setting we take or where is a matrix of dimension 2 × 2 and , , ,  as in (58).
Working as in the doubling iterative algorithm of Section 2.2.1 and using zero initial condition  0 = 0, so  0 =  0  −1 0 = 0; we are able to derive the following nonlinear iterative equations: with initial conditions It is clear that   =  2   −1 2  =  2  tends to a steady state value K.

Algebraic Algorithm for K Computation.
In this section, we present an algebraic algorithm for K computation.Working as in the algebraic algorithm of Section 2.2.2 and using the parameters , , ,  by (58), we derive which is a matrix of dimension 2 × 2.
Then, the steady state Kalman filter is Per-step iterative algorithm 1 Per-step iterative algorithm 1 (71) It is obvious from (71) that the parameters of the steady state Kalman filter are related to the steady state Kalman filter gain.
In particular, the steady state prediction error covariance can be computed via the steady state gain and is given by

Table 1 :
Algorithms for the computation of the steady state Kalman filter gain K.
) 2.4.Advantages of the Proposed Algorithms.All algorithms for the computation of the steady state Kalman filter gain K, presented in Section 2, are summarized in Table 1.It is clear that the direct computation of the Kalman filter gain